Could anyone please define that steps that are necessary to signing an applet, or point me in the right direction? Also, I'm developing the applet on in the Eclipse IDE, so if there is a way to sign applets within eclipse, that would be fantastic .

You need to get a signature ID from one of the various companies that
distribute them. What this gives you is a private key and a public
key. You sign your applets with the private key and give out your
public key so that everyone else can decrypt your private key.
Attached is a URL for one such company:

brief steps for signing a java applet:
1) you develop applet
2) get signature ID from above URL
3) use private key to sign your applet
4) clients get your public key and enter it in their database
5) run applet

To generate a key, use the below java command from command prompt/console

keytool -genkey -alias ujjwal -keyalg -RSA

After this, you will need a digital certificate, if you are looking for free certificates then you will find many site offering free certificates on google.

Note that for development purposes you can also generate your own certificate (step 2 in Ujjwal's list) instead of getting an official one. That's done using the "keytool" executable that comes with the JDK; the process is described in the articles linked in that FAQ page.